91Ó°ÊÓ

The cross product of two vectors is a fundamental operation in vector algebra. It results in a vector that is perpendicular to the plane formed by the initial vectors. The formula for the cross product is:

\[ \text{Torque} \rightarrow \tau = \textbf{r} \times \textbf{F} \]

This means you multiply two vectors, the position vector (\textbf{r}) and the force vector (\textbf{F}), using a specific operation that determines the direction and magnitude of the resulting torque vector.

For example, if you have vectors \textbf{A} = \[a_1 \textbf{i} + a_2 \textbf{j} + a_3 \textbf{k}\] and \textbf{B} = \[b_1 \textbf{i} + b_2 \textbf{j} + b_3 \textbf{k}\], their cross product \textbf{A} × \textbf{B} is calculated as follows:
\[ \textbf{A} \times \textbf{B} = \begin{vmatrix} \textbf{i} & \textbf{j} & \textbf{k} \ a_1 & a_2 & a_3 \ b_1 & b_2 & b_3 \right) \].
This determinant calculation will give you a vector that indicates the direction and magnitude of the torque.

A position vector is a vector that extends from the origin of a coordinate system to the position of a point in space. It is typically denoted by \(\textbf{r}\) and expressed in the form:

\[ \textbf{r} = x \textbf{i} + y \textbf{j} + z \textbf{k} \]

In our problem, the position vector (\textbf{r}) of the jar of jalapeño peppers is given as:
\[ \textbf{r} = 3.0 \text{ m} \textbf{i} - 2.0 \text{ m} \textbf{j} + 4.0 \text{ m} \textbf{k} \]

This vector points from the origin to the coordinates (3.0 m, -2.0 m, 4.0 m). The position vector is crucial in determining the torque because it defines the relative location of the point where the force applies.

A force vector represents the force applied to an object, defined by its magnitude and direction. It is typically expressed in vector form as:

\[ \textbf{F} = F_x \textbf{i} + F_y \textbf{j} + F_z \textbf{k} \]

For example, for the given forces in the problem, we have:
1. Force \(\textbf{F}_A\): \[ \textbf{F}_A = 3.0 \text{ N} \textbf{i} - 4.0 \text{ N} \textbf{j} + 5.0 \text{ N} \textbf{k} \]
2. Force \(\textbf{F}_B\): \[ \textbf{F}_B = -3.0 \text{ N} \textbf{i} - 4.0 \text{ N} \textbf{j} - 5.0 \text{ N} \textbf{k} \]

The direction in which the force vector points determines how the object will move or rotate, while the magnitude (denoted by the values along \textbf{i}, \textbf{j}, and \textbf{k}) determines the strength of the force. These vectors are essential in calculating the torque using the cross product.